From [1]}, [2]}, [3]}, [4]}, we get the least energy solutions to (REF ).
Note that (REF ) has semi-trivial solutions \((\mu ^{-\frac{1}{2}}_1 U_{\varepsilon }, 0)\)  and \((0, \mu ^{-\frac{1}{2}}_2 U_{\varepsilon })\) . Here, we are only interested in the nontrivial solutions of (REF ), this is \((\sqrt{k_1}U_{\varepsilon },\sqrt{k_2}U_{\varepsilon })\) , where \(U_\varepsilon \)  is defined in (REF ), \(k_1=\frac{\beta -\mu _2}{\beta ^2-\mu _1\mu _2}\)  and \(k_2=\frac{\beta -\mu _1}{\beta ^2-\mu _1\mu _2}\) . The main results in this aspect are summarized below. Next, we turn to the related limiting elliptic system (REF ).
